Mitigation of rare events in multistable systems driven by correlated noise

We consider rare transitions induced by colored noise excitation in multistable systems. We show that undesirable transitions can be mitigated by a simple time-delay feedback control if the control parameters are judiciously chosen. We devise a parsimonious method for selecting the optimal control parameters, without requiring any Monte Carlo simulations of the system. This method relies on a new nonlinear Fokker-Planck equation whose stationary response distribution is approximated by a rapidly convergent iterative algorithm. In addition, our framework allows us to accurately predict, and subsequently suppress, the modal drift and tail inflation in the controlled stationary distribution. We demonstrate the efficacy of our method on two examples, including an optical laser model perturbed by multiplicative colored noise.

Figure 1

Summary of the program followed in this paper.

Figure 2

Control of SDE (42) with $\sigma =1.2$, $s_{cor}=0.25$, $\widehat{x}=1$, $a=1$, and increasing values of delay $\tau$. The uncontrolled bistable response is shown in (a). The controlled response is shown in (b) for $\tau =0.1$ ($\tilde{\sigma }=1.26$, ${\tilde{s}}_{cor}=0.28$), in (c) for $\tau =0.2$ ($\tilde{\sigma }=1.34$, ${\tilde{s}}_{cor}=0.31$), and in (d) for $\tau =0.4$ ($\tilde{\sigma }=1.55$, ${\tilde{s}}_{cor}=0.42$).

Figure 3

Contour plots of the peak $x$-coordinate for the uncontrolled and the controlled ($a=1$, $\widehat{x}=1$) SDE (42), as predicted by (36) for $M=2$. The uncontrolled case is shown in (a), while the controlled one is shown in (b) for $\tau =0.1$, in (c) for $\tau =0.2$, and in (d) for $\tau =0.4$. For the uncontrolled SDE, the response PDF is bimodal and symmetric around zero; therefore panel (a) shows the absolute value of the $x$-coordinates of the two peaks.

Figure 4

Control of SDE (42) with $a=1$, $s_{cor}=0.2$, $\tau =0.1$, $\sigma =0.8$ in (a), and $\sigma =1.4$ in (b), with $\widehat{x}=1$ (blue curves), and with peak drift cancel (red curves). The $\widehat{x}\left(R\right)$ in order to achieve peak drift canceling is calculated to 0.83 for (a), and 0.48 for (b). Stationary PDF approximation (36) for $M=2$ is plotted against the PDF obtained by Monte Carlo simulations of the respective SDDE, shown in plots by circles.

Figure 5

Control of SDE (42) with $\sigma =1.4$, $s_{cor}=0.2$, $\tau =0.1$, no peak drift control, and increasing gain $a$. Stationary PDF approximation (36) for $M=2$ (solid curves) is plotted against the PDF obtained by Monte Carlo simulations of SDDE (43), shown by circles.

Figure 6

Three regimes of the controlled stationary PDF as a function of control parameters and correlation time of the noise. The figure corresponds to SDE (42) with $\sigma =1$. In (a) $\widehat{x}=1$, and in (b) $\widehat{x}$ is chosen such that the controlled PDF peak is located at $x=1$. In both figures, the region below the blue surface contains the parameter combinations that result in a bimodal PDF, the region between the two surfaces corresponds to unimodal PDFs with inflated tail, while the region above the green surface to unimodal PDFs with no inflated tail.

Figure 7

Bistable potential (57) for the set of parameters (56).

Figure 8

Control of SDE (55) with $\widehat{x}=42$, $a=4$, $\sigma =2$, $s_{cor}=0.02$, and increasing values of delay $\tau$. The uncontrolled bistable response is shown in (a). The controlled response is shown in (b) for $\tau =0.02$ ($\tilde{\sigma }=2.09$, ${\tilde{s}}_{cor}=0.022$), in (c) for $\tau =0.05$ ($\tilde{\sigma }=2.24$, ${\tilde{s}}_{cor}=0.025$), and in (d) for $\tau =0.08$ ($\tilde{\sigma }=2.43$, ${\tilde{s}}_{cor}=0.029$).

Figure 9

Control of SDE (55) with $\sigma =2$, $s_{cor}=0.02$, for $\tau =0.05$ and increasing control gain $a$. Stationary PDF approximation (36) for $M=2$ (solid curves) is plotted against the PDF obtained by Monte Carlo simulations of SDDE (58), shown by circles.